An example of a simplicial complex is Čech complex.
Figure REF  shows a simple illustrative example. The goal is to recover the topological invariants of the space in Figure REF . It is clear that the 0-betti number is one, since there is only one connected component; and the 1-betti number is two, since there are two holes; the higher dimensional betti numbers are zero. The dataset \(X\)  is given by the six sampled points in Figure REF . To construct the Čech complex over \(X\) , we begin with the points in \(X\)  as 0-simplexes. In order to construct the higher dimensional simplexes, we start growing a ball at each point, as in Figure REF ; at this state, the 0-betti number is six, and 1-betti number is zero. By increasing the radius, some of the balls start to overlap each others; for each \(k+1\)  overlapped balls we insert a \(k\) -simplex. Figure REF  shows a collection of 1-simplexes, line segments joining the two points, created by the pairwise overlap of their corresponding balls; what can be clearly seen in this simplicial complex is that it recovers the topological properties of the underlying structure in Figure REF . If we increase the radius further, the three balls at the top begin to overlap each others, hence we can add a 2-simplex—a filled in triangle—as in Figure REF . Therefore, the hole which was created at Figure REF  disappeared by increasing the radius at Figure REF , as a result that topological property is lost; this could eventually happen for the second hole as we continue to increase the radius and add more simplexes. It is important to notice that the topological properties that persist for a longer period, before they disappear, best represent the properties of the underlying structure. This characteristic is the basic principle of the persistent homology method, which was first formalized in [1]}.
